You could use movement subdivision. If you have an object with a bounding box with a height of 50 and it moves 300 pixels in height that frame then you can move it 6 times 50 pixels and check each time if there is a collision.

A small optimization to this technique is to draw a second bounding box from the start position to the end position of the bounding box and check if anything intersects with this. If not you can skip the movement subdivision test and just move it once.